Joshua – a brief introduction

Hi!

I was born in Johannesburg, South Africa, and spent most of my childhood there; however, I have been living in Cape Town for the last eleven years. I completed my undergraduate degree at the University of Cape Town in 2021, where I majored in pure mathematics and physics. In 2022 I enrolled for an Honours degree in pure mathematics at the University of the Western Cape. This is where I met Professor David Holgate, who introduced me to the world of general topology. At UWC, David leads the Topology for Tomorrow (“T4T”) research group. In 2023 I was fortunate to receive a T4T bursary, which allowed me to pursue my Master’s studies under his supervision.

During my Master’s I focused on learning pointfree topology, viewing it as a blend of lattice theory and general topology. I subsequently decided to turn my attention to foundational topics in the area; you can read my thesis online. In 2024 I became very interested in how one can “freely generate” frames over semilattices, or even sets. This is an example of an adjunction between the category of frames and the category of meet-semilattices. The adjunction shows that the category of frames is algebraic. In 2025 I worked on the Kuratowski–Mrówka theorem for frames and locales, and managed to prove it without the Axiom of Choice, although I still used the Law of Excluded Middle. I believe the late Japie Vermeulen gave a constructive proof.

In February 2026 I began my PhD at UWC, in association with AIMS South Africa, continuing for now on the same trajectory in pointfree topology. At the moment I am working on coproducts of frames, and I want to be able to describe them in full. Perhaps a simpler description of the coproduct will yield a simpler proof of the localic Kuratowski–Mrówka theorem. More on this later!